Problem 59 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=2^{x}\end{equation}

Step-by-Step Solution

Answer

The derivative is \( y' = 2^x \ln(2) \).

Step 1: Identify the Function

The given function is an exponential function, defined as \( y = 2^x \), where the base is 2, and the exponent is the variable \( x \).

Step 2: Recall the Derivative Rule for Exponential Functions

For any exponential function of the form \( y = a^x \), where \( a \) is a constant, the derivative of \( y \) with respect to \( x \) is given by \( y' = a^x \ln(a) \). This rule comes from differentiating \( a^x \) with respect to \( x \) and using logarithmic differentiation.

Step 3: Apply the Derivative Rule

Using the rule from Step 2, substitute \( a = 2 \) into the formula. Thus, the derivative of \( y = 2^x \) with respect to \( x \) is \( y' = 2^x \ln(2) \).

Step 4: Simplify the Derivative Expression

Since there are no additional terms in the expression, the simplified derivative remains \( y' = 2^x \ln(2) \).

Key Concepts

Derivative

Derivatives are a core concept in calculus. They represent the rate at which a function is changing at any given point. For functions like \( y = 2^x \), the derivative tells us how quickly the value of \( y \) increases as \( x \) changes.

• In mathematical terms, if \( f(x) \) is a function, then its derivative, usually denoted as \( f'(x) \), gives us the slope of the tangent line to the curve at any point \( x \).
• The derivative is essential in various fields to model change, like velocity in physics or growth in biology.

To find the derivative of an exponential function, we often use special rules that simplify the process and provide accurate results.

Logarithmic Differentiation

When dealing with complex equations, logarithmic differentiation can be a lifesaver. This method involves taking the natural logarithm of both sides of the equation. It can simplify the differentiation process, especially for exponential and power functions. In our case, the function \( y = 2^x \) is simple enough, but logarithmic differentiation gives us a useful rule:

• We take \( y = a^x \) and write it as \( \ln(y) = x \ln(a) \).
• Differentiating both sides with respect to \( x \) gives us \( \frac{1}{y} \frac{dy}{dx} = \ln(a) \), leading to \( \frac{dy}{dx} = a^x \ln(a) \).

This formula is what provides the derivative of our original function.

Exponential Growth

Exponential functions like \( y = 2^x \) are fundamental in modeling exponential growth. This type of growth is characterized by a constant rate of growth with respect to time. In other words, the more you have, the more you gain. Here's what makes exponential growth unique:

• It increases rapidly, doubling over consistent intervals.
• This is common in natural phenomena, such as populations, radioactive decay, and investments.

Understanding the derivative of an exponential function allows you to predict how quickly a quantity is increasing at any point. As we found in the solution, \( y' = 2^x \ln(2) \), helps determine the exact rate at each moment.

Mathematical Functions

Mathematical functions are like machines that turn an input into an output. They form the heart of mathematical modeling and analysis.
An exponential function is a specific type of mathematical function that describes phenomena where something grows or decays at a constant rate. In an exponential function like \( y = 2^x \):

• The base (2) tells us the factor by which the output multiplies with every unit increase in \( x \).
• The exponent \( x \) is the variable, representing the input upon which the function operates.

Functions are crucial for understanding relationships between numbers, and they allow us to predict future trends based on current data. Having a strong grasp of mathematical functions, including their derivatives, is vital for solving real-world problems.